# Jost Solution and the Spectrum of the Discrete Dirac Systems

- Elgiz Bairamov
^{1}Email author, - Yelda Aygar
^{1}and - Murat Olgun
^{1}

**2010**:306571

**DOI: **10.1155/2010/306571

© Elgiz Bairamov et al. 2010

**Received: **14 September 2010

**Accepted: **10 November 2010

**Published: **29 November 2010

## Abstract

We find polynomial-type Jost solution of the self-adjoint discrete Dirac systems. Then we investigate analytical properties and asymptotic behaviour of the Jost solution. Using the Weyl compact perturbation theorem, we prove that discrete Dirac system has the continuous spectrum filling the segment [-2,2]. We also study the eigenvalues of the Dirac system. In particular, we prove that the Dirac system has a finite number of simple real eigenvalues.

## 1. Introduction

holds [1, chapter 3].

hold. The collection of quantities that specify to as the behaviour of the radial wave functions and at infinity is called the scattering of the BVP (1.1) and (1.2).

where and are real-valued continuous functions. In the case , , where is a potential function and the mass of a particle, (1.11) is called stationary Dirac system in relativistic quantum theory [5, chapter 7]. Jost solution and the scattering theory of (1.11) have been investigated in [6].

Jost solutions of quadratic pencil of Schrödinger, Klein-Gordon, and -Sturm-Liouville equations have been obtained in [7–9]. In [10–17], using the analytical properties of Jost functions, the spectral analysis of differential and difference equations has been investigated.

Discrete boundary value problems have been intensively studied in the last decade. The modelling of certain linear and nonlinear problems from economics, optimal control theory, and other areas of study has led to the rapid development of the theory of difference equations. Also the spectral analysis of the difference equations has been treated by various authors in connection with the classical moment problem (see the monographs of Agarwal [18], Agarwal and Wong [19], Kelley and Peterson [20], and the references therein). The spectral theory of the difference equations has also been applied to the solution of classes of nonlinear discrete Korteveg-de Vries equations and Toda lattices [21].

In this paper, we find Jost solution of (1.12) and investigate analytical properties and asymptotic behaviour of the Jost solution. We also show that, , where denotes the continuous spectrum of , generated in by (1.12) and (1.13).

We also prove that under the condition (1.14) the operator has a finite number of simple real eigenvalues.

## 2. Jost Solution of (1.12)

Theorem 2.1.

Proof.

By the condition (1.14), the series in the definition of ( ) are absolutely convergent. Therefore, ( ) can, by uniquely be defined by and , that is, the system (1.12) for and , has the solution given by (2.4) and (2.5).

where is the integer part of and is a constant. It follows from (2.4) and (2.11) that (2.3) holds.

Theorem 2.2.

The solution has an analytic continuation from to .

Proof.

From (1.14) and (2.11), we obtain that the series and are uniformly convergent in . This shows that the solution has an analytic continuation from to .

The functions and are called Jost solution and Jost function of the BVP (1.12) and (1.13), respectively. It follows from Theorem 2.2 that Jost solution and Jost function are analytic in and continuous on .

Theorem 2.3.

Proof.

From (2.15) and (2.16), we obtain (2.12).

## 3. Continuous and Discrete Spectrum of the BVP (1.12) and (1.13)

Theorem 3.1.

Proof.

where denotes the set of all eigenvalues of .

Definition 3.2.

The multiplicity of a zero of the function is called the multiplicity of the corresponding eigenvalue of .

Theorem 3.3.

Under the condition (1.14), the operator has a finite number of simple real eigenvalues.

Proof.

To prove the theorem, we have to show that the function has a finite number of simple zeros.

that is, all zeros of are simple.

There is a contradiction comparing (3.16) and (3.18). So and function has only a finite number of zeros.

## Authors’ Affiliations

## References

- Marchenko VA:
*Sturm-Liouville Operators and Applications, Operator Theory: Advances and Applications*.*Volume 22*. Birkhäuser, Basel, Switzerland; 1986:xii+367.View ArticleGoogle Scholar - Levitan BM:
*Inverse Sturm-Liouville Problems*. VSP, Zeist, The Netherlands; 1987:x+240.MATHGoogle Scholar - Chadan K, Sabatier PC:
*Inverse Problems in Quantum Scattering Theory*. Springer, New York, NY, USA; 1977:xxii+344.View ArticleMATHGoogle Scholar - Zaharov VE, Manakov SV, Novikov SP, Pitaevskiĭ LP:
*Theory of Solutions*. Plenum Press, New York, NY, USA; 1984.Google Scholar - Levitan BM, Sargsjan IS:
*Sturm-Liouville and Dirac Operators, Mathematics and Its Applications*.*Volume 59*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1991:xii+350.View ArticleGoogle Scholar - Gasymov MG, Levitan BM:
**Determination of the Dirac system from the scattering phase.***Doklady Akademii Nauk SSSR*1966,**167:**1219-1222.MathSciNetMATHGoogle Scholar - Jaulent M, Jean C:
**The inverse**s**-wave scattering problem for a class of potentials depending on energy.***Communications in Mathematical Physics*1972,**28:**177-220. 10.1007/BF01645775MathSciNetView ArticleGoogle Scholar - Bairamov E, Karaman Ö:
**Spectral singularities of Klein-Gordon**s**-wave equations with an integral boundary condition.***Acta Mathematica Hungarica*2002,**97**(1-2):121-131.MathSciNetView ArticleMATHGoogle Scholar - Adıvar M, Bohner M:
**Spectral analysis of**q**-difference equations with spectral singularities.***Mathematical and Computer Modelling*2006,**43**(7-8):695-703. 10.1016/j.mcm.2005.04.014MathSciNetView ArticleMATHGoogle Scholar - Krall AM:
**A nonhomogeneous eigenfunction expansion.***Transactions of the American Mathematical Society*1965,**117:**352-361.MathSciNetView ArticleMATHGoogle Scholar - Lyance VE:
**A differential operator with spectral singularities, I.***AMS Translations*1967,**2**(60):185-225.Google Scholar - Lyance VE:
**A differential operator with spectral singularities, II.***AMS Translations*1967,**2**(60):227-283.Google Scholar - Bairamov E, Çelebi AO:
**Spectrum and spectral expansion for the non-selfadjoint discrete Dirac operators.***The Quarterly Journal of Mathematics. Oxford Second Series*1999,**50**(200):371-384. 10.1093/qjmath/50.200.371MathSciNetView ArticleMATHGoogle Scholar - Krall AM, Bairamov E, Çakar Ö:
**Spectrum and spectral singularities of a quadratic pencil of a Schrödinger operator with a general boundary condition.***Journal of Differential Equations*1999,**151**(2):252-267. 10.1006/jdeq.1998.3519MathSciNetView ArticleMATHGoogle Scholar - Bairamov E, Çakar Ö, Krall AM:
**An eigenfunction expansion for a quadratic pencil of a Schrödinger operator with spectral singularities.***Journal of Differential Equations*1999,**151**(2):268-289. 10.1006/jdeq.1998.3518MathSciNetView ArticleMATHGoogle Scholar - Bairamov E, Çakar Ö, Krall AM:
**Non-selfadjoint difference operators and Jacobi matrices with spectral singularities.***Mathematische Nachrichten*2001,**229:**5-14. 10.1002/1522-2616(200109)229:1<5::AID-MANA5>3.0.CO;2-CMathSciNetView ArticleMATHGoogle Scholar - Krall AM, Bairamov E, Çakar Ö:
**Spectral analysis of non-selfadjoint discrete Schrödinger operators with spectral singularities.***Mathematische Nachrichten*2001,**231:**89-104. 10.1002/1522-2616(200111)231:1<89::AID-MANA89>3.0.CO;2-YMathSciNetView ArticleMATHGoogle Scholar - Agarwal RP:
*Difference Equations and Inequalities: Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 228*. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.Google Scholar - Agarwal RP, Wong PYJ:
*Advanced Topics in Difference Equations*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997.View ArticleMATHGoogle Scholar - Kelley WG, Peterson AC:
*Difference Equations: An Introduction with Applications*. 2nd edition. Harcourt/Academic Press, San Diego, Calif, USA; 2001:x+403.MATHGoogle Scholar - Toda M:
*Theory of Nonlinear Lattices, Springer Series in Solid-State Sciences*.*Volume 20*. Springer, Berlin, Germany; 1981:x+205.Google Scholar - Glazman IM:
*Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators*. Israel Program for Scientific Translations, Jerusalem, Palestine; 1966:ix+234.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.